Problem: What integer $n$ satisfies $0\le n<19$ and $$-200\equiv n\pmod{19}~?$$
Noticing that $190\equiv0\pmod{19}$ and $-200+190=-10$, we can say that  \[-200\equiv n\pmod{19}\]if and only if  \[-10\equiv n\pmod{19}.\]This is not in the range $0\leq n<19$, but adding 19 again gives \[9\equiv n\pmod{19}.\]The answer is $n=\boxed{9}$.